Find the equation of the set of points P, the sum of whose distances from A \left(\right. 4 , 0 , 0 \left.\right) and B \left(\right.

Engineering

Mathematics

Line of Greatest Slope

Question

Find the equation of the set of points P, the sum of whose distances from

A (4, 0, 0)

and

B (- 4, 0, 0)

is equal to 10.

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Let

(x, y, z)

be a point with the sum of whose distance from

(4, 0, 0)

and

(- 4, 0, 0)

is equal to 10. By using the distance formula between two points we have

\sqrt{(x - 4)^{2} + (y - 0)^{2} + (z - 0)^{2}} + \sqrt{(x + 4)^{2} + (y - 0)^{2} + (z - 0)^{2}} = 10

By rearranging and squaring we get

(x - 4)^{2} + y^{2} + z^{2} = 100 + (x + 4)^{2} + y^{2} + z^{2} - 20 \sqrt{(x + 4)^{2} + y^{2} + z^{2}}

By cancellation,

20 \sqrt{(x + 4)^{2} + y^{2} + z^{2}} = 16 x + 100

Dividing by 4,

5 \sqrt{(x + 4)^{2} + y^{2} + z^{2}} = 4 x + 25

Square both sides,

25 (x^{2} + 8 x + 16 + y^{2} + z^{2}) = 16 x^{2} + 200 x + 625

\Rightarrow 9 x^{2} + 25 y^{2} + 25 z^{2} = 225

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